$ C = \left[\begin{array}{rrr}5 & 0 & -2 \\ 5 & -1 & 2\end{array}\right]$ $ w = \left[\begin{array}{r}3 \\ 0 \\ 5\end{array}\right]$ What is $ C w$ ?
Because $ C$ has dimensions $(2\times3)$ and $ w$ has dimensions $(3\times1)$ , the answer matrix will have dimensions $(2\times1)$ $ C w = \left[\begin{array}{rrr}{5} & {0} & {-2} \\ {5} & {-1} & {2}\end{array}\right] \left[\begin{array}{r}{3} \\ {0} \\ {5}\end{array}\right] = \left[\begin{array}{r}? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{5}\cdot{3}+{0}\cdot{0}+{-2}\cdot{5} \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{5}\cdot{3}+{0}\cdot{0}+{-2}\cdot{5} \\ {5}\cdot{3}+{-1}\cdot{0}+{2}\cdot{5}\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{5}\cdot{3}+{0}\cdot{0}+{-2}\cdot{5} \\ {5}\cdot{3}+{-1}\cdot{0}+{2}\cdot{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}5 \\ 25\end{array}\right] $